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I meanwhile found that in the ring of multivariate power series the above product can be truncated using bigoh with two arguments:
sage: R.<a,b> = PowerSeriesRing(QQ); R
Multivariate Power Series Ring in a, b over Rational Field
sage: (1+a+a^2+a^3)*(1+b+b^2+b^3)+O(a,b)^4
1 + a + b + a^2 + a*b + b^2 + a^3 + a^2*b + a*b^2 + b^3 + O(a, b)^4
